APPENDIX TO CHAPTER IX

APPLICATION OF ANALYTIC FUNCTIONS TO PLANE TOPOLOGY

(Eilenberg's Method)

1. INDEX OF A POINT WITH RESPECT TO A LOOP

(Ap.1.1) If t -> y(f) (a^t^b) is a path in an open subset A of C, there is
in A a homotopy cpofy into a road yl5 such that <p is defined in [a, b] x [0, 1]
and <p(a, $) = y(a) and (p(b, £) = y(b)for every £ e [0, 1].

Let I = [a, b]; asy(I) is compact, d(y(l\ C - A) = p is > 0 (3.17.11). Asy
is uniformly continuous in I (3.16.5), there is a strictly increasing sequence
(tk)o^k**m of points of I such that t0 = a,tm = b, and the oscillation (Section
3.14) of y in each of the intervals [tk, tk+1] (Q ^k^m — 1) is < p. Define

y, in I as follows: for tk ^ t^ tk+1, y^t) = y(tk) + / ~ ** (y('fc+i) - ?(**))

*Jt + l ~~ lk

(0 < /fc < m — 1); it is clear that y^ is a road, with y^a) = y(a\ y^b) = y(b),
and y^l) is contained in A, since yi([tk9 tk+l]) is contained in the open ball
of center y(tk) and radius p. Define then cp(t, £) = Zy^t) + (1 — fyy(t); it
is readily verified that cp(t9 f) is in the open ball of center y(tk) and radius p
for tk ^ t ^ tk+l and 0 ^ £, < 1 (0 < fc < m — 1); hence q> verifies the required
conditions.

In particular, if 7 is a loop, we see that cp is a loop homotopy in A of y
into a circuit y^

Consider now any loop y in C, defined in I, and any point a $ y(I). As
there are, by (Ap.1.1), circuits yl which are homotopic to y in C — {a},
we can define the index j(a; y) as equal to j(a; yt) for any circuit homotopic
to y in C - {a}; by Cauchy's theorem (9.6.3), this is independent of the
particular circuit yt homotopic to y in C — {a}.

251eighborhood of H> — 0